Question

If x is a real number, prove that -X<=|x|​

Answer 1

Answer:

Step-by-step explantion:

Definition of absolute value of x:

|x|=x assuming x has taken a positive value.

|x|=-x assuming x has taken a negative value.

|x|=0 assuming x is 0.

----------------------------------------------------------------------------

$-x \le |x|$

Add $x$ on both sides:

$0 \le |x|+x$

$|x|+x \ge 0$

Let $h(x)=|x|+x$.

Assume x is positive, then $h(x)=x+x=2x$ where $2x$ is positive since $x$ is positive. Also $h(x)=2x>0$ for positive x.

Assume $x$ is negative, then $h(x)=-x+x=0$.

Assume $x=0$, then $h(0)=|0|+0=0+0=0$.

The result for h is positive when x is positive but 0 elsewhere for x.

-------------------------------------End of Proof--------------------------------------